We study the structure and viscoelastic behavior of 3D plastic crystals of colloidal dumbbells in an oscillatory shear field based on a combination of small-angle neutron scattering experiments under shear (rheo-SANS) and Brownian dynamics computer simulations. Sterically stabilized dumbbell-shaped microgels are used as hard dumbbell model systems which consist of dumbbell-shaped polystyrene (PS) cores and thermosensitive poly(N-isopropylacrylamide) (PNIPAM) shells. Under increasing shear strain, a discontinuous transition is found from a twinned-fcc-like crystal to a partially oriented sliding-layer phase with a shear-molten state in between. In the novel partially oriented sliding-layer phase, the hard dumbbells exhibit a small but finite orientational order in the shear direction. We find that this weak correlation is sufficient to perturb the nature of the nonequilibrium phase transition as known for hard sphere systems. The discontinuous transition for hard dumbbells is observed to be accompanied by a novel yielding process with two yielding events in its viscoelastic shear response, while only a single yielding event is observed for sheared hard spheres. Our findings will be useful in interpreting the shear response of anisotropic colloidal systems and in generating novel colloidal crystals from anisotropic systems with applications in colloidal photonics.